QuestionJuly 15, 2025

33. Use properties of logarithms to expand ln(2bsqrt ((b+1)/(b-1)))

33. Use properties of logarithms to expand ln(2bsqrt ((b+1)/(b-1)))
33. Use properties of logarithms to expand ln(2bsqrt ((b+1)/(b-1)))

Solution
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Answer

\ln(2) + \ln(b) + \frac{1}{2}[\ln(b+1) - \ln(b-1)] Explanation 1. Apply the logarithm of a product Use **\ln(xy) = \ln(x) + \ln(y)** to expand \ln(2b\sqrt{\frac{b+1}{b-1}}) into \ln(2b) + \ln(\sqrt{\frac{b+1}{b-1}}). 2. Expand \ln(2b) Use **\ln(xy) = \ln(x) + \ln(y)** again to expand \ln(2b) into \ln(2) + \ln(b). 3. Simplify the square root term Use **\ln(\sqrt{x}) = \frac{1}{2}\ln(x)** to simplify \ln(\sqrt{\frac{b+1}{b-1}}) into \frac{1}{2}\ln(\frac{b+1}{b-1}). 4. Apply the logarithm of a quotient Use **\ln(\frac{x}{y}) = \ln(x) - \ln(y)** to expand \frac{1}{2}\ln(\frac{b+1}{b-1}) into \frac{1}{2}[\ln(b+1) - \ln(b-1)].

Explanation

1. Apply the logarithm of a product<br /> Use **$\ln(xy) = \ln(x) + \ln(y)$** to expand $\ln(2b\sqrt{\frac{b+1}{b-1}})$ into $\ln(2b) + \ln(\sqrt{\frac{b+1}{b-1}})$.<br />2. Expand $\ln(2b)$<br /> Use **$\ln(xy) = \ln(x) + \ln(y)$** again to expand $\ln(2b)$ into $\ln(2) + \ln(b)$.<br />3. Simplify the square root term<br /> Use **$\ln(\sqrt{x}) = \frac{1}{2}\ln(x)$** to simplify $\ln(\sqrt{\frac{b+1}{b-1}})$ into $\frac{1}{2}\ln(\frac{b+1}{b-1})$.<br />4. Apply the logarithm of a quotient<br /> Use **$\ln(\frac{x}{y}) = \ln(x) - \ln(y)$** to expand $\frac{1}{2}\ln(\frac{b+1}{b-1})$ into $\frac{1}{2}[\ln(b+1) - \ln(b-1)]$.
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